Variation has played a large role in science and art. Scientists have spent much of their time explaining the changing nature of countless aspects of the world and its universe. To create variations in systems under study or design, scientists and engineers have had to think through the desired variations and enact them by hand. In recent years, computers have aided this process, by making the enactment process faster. For instance, an engineer could first simulate a design which had been changed from the original, thus testing it before having to spend money building something which might not be as good as the original. But the changes, or variations, in that design would first have to be conceived or modeled by the engineer.
Similarly, musical variations occur because the artist has created them, either by hand, or with the aid of computer programs. The computer may introduce elements of randomness or use tightly (or loosely) controlled parameters to add extra components to the work at hand. The methods employed, however, are often narrow in scope, having been designed by and for individuals and their respective projects. These earlier approaches do not accommodate the disparate styles of composers today. As a simple example, consider an opening and closing filter used to change the timbre of a sound collage. This provides variations on the original sound piece, but it is not suitable for a wide range of musical taste.
The technique proposed in accordance with the present invention, however, generates variations for music of any style, making it a versatile tool for composers wishing to develop their musical material. There is no limit on the number of variations possible. The variations can closely mirror the original work, diverge substantially, or retain some semblance of the source piece, and are created through the use of a mathematical concept, later more fully explained and referenced, involving the mapping of so-called "chaotic" trajectories successively displaced from one another.